I have been running missions for my faction in a system where they are the second placed faction with regards to influence and have been at it for a couple of weeks, I have kept a watch on the sloooooow progress of the influence percentage rising (when I started they were 22% and the controlling faction 59%) on Friday it was up to 28% but I logged in last night (the following day, Saturday) and had a look and my factions influence was down to 22.5%!..
I am rather perplexed at this and am hoping someone with more knowledge than I can explain what happened
My little birds whisper to me the strangest stories...
That in the system where this is happening the controlling faction you are trying to overturn is called "Left Party of Cowiskimo", which while on the face of it looks like just any other NPC faction, they actually appear to have quite a considerable CMDR backing for whatever reason (whether organised groups or multiple randoms working for them) as can be deduced by the fact that this faction has been
spread to 24 systems
My own faction has a similar annoying NPC faction like this who are present in 35 systems on our doorstep, including in a few where our faction is present called Party Of Yoru, which because getting allied status with them is a prerequisite for unlocking one of the main engineers (Zach Nemo), they will have 100's of cmdrs working for them at any one time, which is how they managed to expand to 35 systems and means we often get randoms coming in to the few systems we share with POY and working against us for them.
It looks like it might be similar thing happening to you with Left Party of Cowiskimo, if they are not a prereq for one of the engineers could be getting allied with them unlocks a system permit somewhere, or else a player group has adopted this NPC faction perhaps, must be some reason for it.